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Optimization and Thermal Performance Assessment of Pin-Fin Heat Sinks Z. S. Abdel-Rehim a a National Research Center, Dokki, Giza, Egypt Online Publication Date: 01 January 2009

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Energy Sources, Part A, 31:51–65, 2009 Copyright © Taylor & Francis Group, LLC ISSN: 1556-7036 print/1556-7230 online DOI: 10.1080/15567030701468118

Optimization and Thermal Performance Assessment of Pin-Fin Heat Sinks Z. S. ABDEL-REHIM1

Downloaded By: [Abdel-Rehim, Z. S.] At: 08:52 30 November 2008

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National Research Center, Dokki, Giza, Egypt Abstract In this work, the heat transfer and fluid flow analyses are employed to optimize the geometry of the pin-fin heat sinks. An entropy generation minimization method is employed to optimize the overall thermal performance and behavior of pinfin heat sinks. The performance of the heat sinks is determined by its thermal resistance and pressure drop, since they significantly influence the thermal resistance during forced convection cooling. The optimum design of heat sink for in-line and staggered alignments with circular, square, rhombus, rectangular, and elliptical configurations are investigated, and the thermal behavior is compared. The entropy generation rate is developed using mass, energy, and entropy balance over the control volume. The formulation for the dimensionless entropy generation rate due to heat transfer and due to fluid flow (pressure drop) is obtained in terms of fin geometry, thermal conductivity, pin-fin alignment, Reynolds, and Prandtl numbers. Using an energy balance equation over the same control volume, the average heat transfer coefficient for heat sink is developed, which is a function of the heat sink material, fluid properties, fin geometry, pin-fin alignment. The selected materials are plastic, aluminum, and copper. Thermal and hydrodynamic analysis of pin-fin heat sink are performed using parametric variations of each design variable, including pin diameter or side, pin height, approach velocity, number of pin-fins, and thermal conductivity of the material. Optimization of heat sink designs and parametric behavior are presented and compared based on the selected pin-fin configurations, alignment, and material property. The results indicate that geometries of circular and elliptical shapes provide more favorable conditions for heat transfer than that of square, rectangular, and rhombus shapes. In all cases, optimum size of staggered alignment is better than in-line alignment. In a particular pin-fin configuration, entropy generation rate is comparatively elliptical < circular < rhombus < rectangular < square pin-fins. The optimum entropy generation rate moves down as the thermal conductivity of the fin material increases. Copper fin gives the best performance, and if the weight of the heat sink is a constraint, the aluminum fin would be preferable. However, the optimum diameter for the selected materials is almost the same. The optimum entropy generation rate moves down again with the thermal conductivity of the fin material, which shows the best performance of copper fins for the same approach velocity. Also, the optimum approach velocity for the selected materials is almost the same. The entropy generation rate decreases with the increase in thermal conductivity for the same number of fins and under the same operating conditions. Also, the optimum fin length increases with thermal conductivity of fin materials. The results enable the designer to quickly and easily access the merits of pin-fin geometries for specific design conditions and for selecting the optimal dimensions of fin and material. Keywords entropy generation minimization, fluid flow, heat transfer, optimization, pin-fin heat sinks, thermal conductivity, thermal performance

Address correspondence to Dr. Zeinab Abdel-Rehim, Mechanical Engineering Department, National Research Center, El-Bohouth St., Dokki, Giza, Egypt. E-mail: [email protected]

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1. Introduction With the increase in heat dissipation from microelectronic devices, the thermal management becomes a very important element of the electronic product design. Heat removal has played an important role in maintaining reliable operation of the electronic devices. The heat transfer, fluid flow, and thermal design are more important than ever, and the development of future large-scale, high-speed circuits may well be limited by inability to maintain effective cooling. Heat sinks are devices that enhance heat dissipation from a hot surface, usually the case of a heat generation component, to a cooler ambient, usually air. Heat sinks are the most common thermal management hardware in use in microelectronics. Pin-fin arrays are widely employed to enhance the heat transfer rate in electronic components. In designing a pin-fin array, the criteria generally adopted is either to maximize the heat transfer rate under a given fin volume (weight) or to minimize the fin volume under a prescribed heat load. An overview of different types of heat sinks and associated design parameters is provided by Lee (1995). Bejan (1996) provides entropy generation minimization (EGM) method which is a procedure for simultaneously assessing the parametric relevance of system parameters as they relate to both thermal performance as well as viscous effects. Culham and Muzychka (2001) provided a procedure that allows the simultaneous optimization of heat sink design parameters based on EGM for plate-fin-type heat sink. Optimal design methodology of plate-fin heat sinks for electronic cooling by applying the entropy generation rate to obtain the highest heat transfer efficiency was recently presented by Shih and Liu (2004). Optimal fin geometry based on exergoeconomic analysis for a pin-fin array with the application to electronic cooling was presented by Shuja (2002). Thermal modeling of isothermal cuboids and rectangular heat sinks cooled by natural convection was presented by Culham et al. (1995). Shaukatullah et al. (1996) performed a study to optimize the design of pin-fin heat sinks for use in low-velocity applications such as in personal computers. They found that the pin-fin heat sink up to 15 mm high and base sizes of about 25 25 mm, the 6 6 pin-fin configuration, with fin cross section of 1.5 1.5 mm, appears to be a good practical choice for use in low-velocity open-flow-type conditions. A semi-empirical zonal approach for the design and optimization of pin-fin heat sinks cooled by impingement was presented by Kondo et al. (2000). They calculated the thermal resistance and pressure drop for an air-cooled heat sink. In this article, the heat-transfer and fluid-flow analyses are employed to optimize the geometry of the pin-fin heat sinks. An EGM method is employed to optimize the overall thermal performance and behavior of pin-fin heat sinks. The performance of the heat sinks is measured by its thermal resistance and pressure drop, since they significantly influence the thermal resistance during forced convection cooling. Three materials are selected: plastic, aluminum, and copper to optimize the material with the fin diameter. The optimum design of heat sink for in-line and staggered alignments with circular, square, rectangular, rhombus, and elliptical cross sections and material property (thermal conductivity) are investigated, and the thermal behavior is compared.

2. Problem Optimization Optimum design creates the best available heat sink solution for the electronic applications. When designing a heat sink, one needs to examine various parameters that affect not only the heat sink performance, but also the overall performance of the system. Several


Pin-Fin Heat Sinks

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independent group variables, such as base plate dimensions, heat source dimensions, heat load, pin-fin configuration and dimensions, pin-fin arrangement (in-line or staggered), material properties (thermal conductivity), and fluid properties, can improve heat sink performance with respect to the selected design criteria. A list of design constraints for a heat sink may include parameters such as minimum chip temperature, minimum pressure drop, minimum size, minimum heat sink mass or weight, induced approach flow velocity, cross-sectional geometry of incoming flow, amount of required heat dissipation, minimum cost, and orientation with respect to the gravity. The parameters over which a designer has a control for optimization include fin height, length, thickness, fin spacing, number of fins, density, fin configuration/profile, base plate thickness, and heat sink material/properties. Most of the previous parameters are interdependent of the others. For example, a larger heat sink surface area will improve cooling, but may increase the weight and cost. Increasing the base plate thickness, which distributes heat more uniformly to the fins of the package, is smaller than the heat sink, but increases the weight. Thicker fins provide more structural integrity and may be easier to manufacture, but increase the weight for a given thermal resistance.

3. Thermal Analysis 3.1.

Fin Temperature

Increase of power densities in microelectronics and simultaneous drive to reduce the size and weight of electronic products have led to the greater importance of thermal management issues in the industry. For a given heat load, Q, and ambient fluid temperature, Ta , the fin temperature, Tf , depends on the total heat sink resistance, Rhs , as: Tf D Q Rhs C Ta ; ı C;

(1)

where Rhs D

Tfa ı 1 D ; C/W; havg Ahs Q

(2)

where havg and Ahs are the average heat transfer coefficient and total surface area of the heat sink, respectively, and Tfa is the average temperature difference between the heat sink (fin) and the ambient fluid. Bahrami et al. (2003, 2004) have presented analytical and empirical models for calculating the thermal resistance under different conditions. Equation (2) shows that the heat sink resistance can be decreased by increasing havg and/or by increasing Ahs ; havg depends on the geometry of the heat sink in a complex manner of the following equation, Bejan (1996): havg D C

Kf 1=2 1=3 ReD Pr ; D

(3)

where C is a constant depending upon the geometry and, for both staggered and inline alignments, can be determined as 1.357 for rectangular, 5.781 for circular, 5.783 for elliptical (Morgan, 1975; Perkins and Leppert, 1964), 0.0 for square, and 0.01 for rhombus.

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3.2. Entropy Generation Rate in Heat Sink Entropy generation rate can be obtained by combining mass, force, energy, and entropy balance across the heat sink. EGM methods will be developed for optimizing the overall performance associated with the following features: 1. Single pin-fin of selected cross-sectional configuration; circular, square, rectangular, rhombus, and elliptical. 2. Alignment; in-line and staggered.


4. EGM Method for Optimization of Fin Geometry Extended surface (fins) constitute one of the most effective design features for promoting heat transfer between solid surface and stream of fluid. This approach consists of calculating the entropy generation rate of fin array and minimizing it systematically. First and second laws of thermodynamics for the control volume are taken together for steady state, and the entropy generated by any engineering system is directly proportional to the work lost irreversibly by the system. This fact is expressed concisely as follows: X

all system components

WLost SPgen D ; W=ı C; Ta

(4)

where SPgen is the entropy generated in each component of the system, WLost is the variable work, and Ta the absolute value of ambient temperature. There are heat-transfer and fluid-friction irreversibility contributions in case of forced convection through the heat sinks. The entropy generation rate balance can be written as: SPgen D .SPgen /T C .SPgen /P ;

(5)

where the term .SPgen /T is the entropy generation due to heat transfer, and the term .SPgen /P is the entropy generation due to pressure drop. The derivation of Eq. (4) as given by Shuja (2002) is used in this study. Where the term .SPgen /T can be expressed as: s s ( " ! #) 1 p p K K K Re f f L f .SPgen /T D ReD NV N u tanh 2 Nu C Nu G1 2 ks Ks ReD Ks (6) where is the parameter which is the function of maximum average fluid velocity, Umax , thermal conductivity, Ks , of fin and the kinematics viscosity, of fluid (air), and can be expressed as: D

Umax qb2 ; 2 Ks T1

(7)

and the term .SPgen /P can be expressed as: .SPgen /P D

N ˇ ReL ReD CD G2 2

(8)

Pin-Fin Heat Sinks

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where ˇ is the fluid-fraction irreversibility coefficient which is a function of the fluid density, , and kinematics viscosity, of fin, and can be expressed as: ˇD

3 T1 Ks ; qb2

(9)


where T1 is the absolute temperature of free stream and qb is the base heat transfer. G1 and G2 (in Eqs. (5) and (7)) are dimensionless geometric parameters which are expressed as the following equations: Sp Sn Sn G1 D .N 1/ C 1 .V a/ C1 N V; (10) Sn D D 4 where D is pin-fin diameter and Sn and Sp are pin spacing in a span-wise direction and stream-wise direction, respectively. Sn G2 D .V a/ 1 ; (11) D where a is fin array geometric parameter and, in this study, is taken as 1 for in-line and 1/2 for staggered, and N and V are the numbers of rows and columns in the fin array, respectively. N u and CD are the Nusselt number and the drag coefficient, respectively, and the following function forms for them with the constants, Reynolds and Prandtl numbers, which were obtained from Shuja (2002) and Kothandaraman and Subramanyan (1989). CD D Cm Rem D;

(12) 1

N u D Cn RenD Pr3 ; where Cm , Cn , m, and n are constants. By introducing the dimensionless parameter D

(13) r

Ks Kf 1 Pr6

, where Ks and Kf are the

thermal conductivities of fin and fluid, respectively, Eq. (4) can be rewritten as the following: p p n C1 Cn n2 1 nC1 SPgen D 2 N V Cn ReD2 tanh 2 ReD ReL C Cn ReD G1 2 M C

N mC1 ˇCm ReD ReL G2 2

1

(14)

The fin length or diameter is expressed in terms of ReL and ReD , respectively.

5. Problem Formulation and Parametric Study The simplest approach to entropy EGM is obtained by fixing all variables in the heat sink design except one, and then monitoring the change in the entropy generation as that particular design variables change over the typical range. Multivariable Newton Raphson

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Z. S. Abdel-Rehim Table 1 Quantities of the parameters of pin-fin geometry and physics and thermal properties of pin-fin material and fluid (air)


Parameters Height, mm Pin diameter, mm Base plate thickness, mm Perimeter, mm Number of pins (in-line) Number of pins (staggered) Overall height of heat sink, mm Approach velocity, m/sec Thermal conductivity of aluminum, W/m.K Thermal conductivity of copper, W/m.K Thermal conductivity of plastics, W/m.K Thermal conductivity of air, W/m.K Specific heat capacity of air, kJ/kg.K Density of air, kg/m3 Kinematics viscosity, m2 /sec Prandtl number (air) Ambient temperature, ı C Heat load, W

Quantizes 50 2 2 15 10 10 11 9 12 1 237 400 25 0.026 1.007 1.1614 1:58 10 0.71 27 10

5

numerical method, Culham et al. (2001), for nonlinear unconstrained optimization using an in-house FORTRAN code to a set of typical design parameters found in electronic applications, is applied to solve this problem. In this case, it is assumed that the following parameters are fixed: the thermal conductivity of fin .Ks / D 237 W/mK, ambient fluid S (air) temperature (Ta ) D 300 K, the spacing between the fins are taken as Spn D 1 and Sn D 1:65, and maximum average fluid velocity, Umax D 1 m/sec. Also, it is assumed D that the total heat dissipation, qb , of 0.5 W is uniformly applied over the base plate of the heat sink, N V D 10 10, and heat load D 10 W. Five different cases of both in-line and staggered alignments are illustrated. These cases are circular pin-fin, square pin-fin, rhombus pin-fin, rectangular pin-fin, and elliptical pin-fin. Table 1 gives the quantities of the parameters of pin-fin geometry and physics and thermal properties of pin-fin material and fluid (air) at ambient temperature. These quantities are used as the default case to compare the performance of each geometry that was selected in this study.

6. Results and Discussion The parametric behavior, responding to a chosen variable with all others unchanged, is studied. Due to the lack of space, only a few calculations are presented and discussed. The result of the thermal resistance with temperature difference between the fin and the ambient is shown in Figure 1. It is noticed that, as the temperature difference increases, the thermal resistance increases. Figure 2 shows the resulting of sink-to-ambient thermal resistance, Rhs , as a function of the number of fins. As the number of fins increases,

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Figure 1. Effect of the temperature difference between fin and ambient on the thermal performance behavior.

the total convective surface area increases. However, as the number of fins goes on to increase, the net effect reverses, and the performance decreases beyond the optimum number of fins. As expected, Figure 2 reveals the existence of an optimum number of fins. The recommended number of fins should be slightly less than 10 fins. Figure 3 shows the parametric performance as a function of fin length. Initially, the performance improves as the total convective surface area increases with fin height. Therefore, “the rate of return on investment” diminishes as the height becomes longer. This is due to temperature rise in the air stream between fin surfaces. Figures 4 and 5 show the balance between entropy generation due to heat transfer and entropy generation due to fluid flow, with increasing the fin diameter in case of circular pin-fin for staggered and in-line, respectively. It is clear that, as the diameter increases, the entropy generated due to heat transfer decreases, while the entropy generated due to fluid flow increases according to the balance between them. Figures 5–9 illustrate the effect of number of fins on the entropy generation rate for staggered and in-line alignments for the selected different geometry configurations: circular, square, rhombus, rectangular, and elliptical. From these figures, the optimum number of fins for both staggered and in-line alignment is determined.

Figure 2. Effect of number of fins on the thermal performance behavior.

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Figure 3. Effect of fin height on the thermal performance behavior.

Figure 4. Effect of fin diameter on entropy generation (heat transfer and fluid flow) for staggered alignment (circular pin-fin).

Figure 5. Effect of number of fins on entropy generation rate for staggered and in-line alignment (circular pin-fin).


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Figure 6. Effect of number of fins on entropy generation rate for staggered and in-line alignment (square pin-fin).

Figure 7. Effect of number of fins on entropy generation rate for staggered and in-line alignment (rhombus pin-fin).

Figure 8. Effect of number of fins on entropy generation rate for staggered and in-line alignment (rectangular pin-fin).

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Figure 9. Effect of number of fins on entropy generation rate for staggered and in-line alignment (elliptical pin-fin).

The comparisons between the different configurations of pin-fin—circular, square, rhombus, rectangular, and elliptical—cross-section configuration, considering the EGM and diameter or side for a given example (qb D 0:5 W, N V D 10 10, D 100, S T1 D 330 K, Spn D 1 and SDn D 1:65) for in-line and staggered alignments, are shown in Figures 10 and 11, respectively. It is very clear that for a particular pin-fin configuration, entropy generation rate is comparative as elliptical < circular < rhombus < rectangular < square pin-fins. Figure 12 depicts the comparison between the entropy generation rate with velocity for both staggered and in-line alignment. It is noticed that, with the increase in the velocity, there is increase in the total entropy generation rate. When the fluid velocity increases, there is increase in the entropy generation due to fluid flow also, due to increase in the pressure drop. Hence, staggered alignment results in higher total entropy generation rate than in-line alignment. The variation of entropy generation rate with pinfin diameter for design variables: in-line alignment, circular pin-fin configuration, and the selected fin materials—plastic, aluminum, and copper—is shown in Figure 13. It is

Figure 10. Entropy generation rate vs. diameter of the selected configurations for in-line alignment.


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Figure 11. Entropy generation rate vs. diameter of the selected configurations for staggered alignment.

Figure 12. Entropy generation rate vs. velocity of the selected configurations for in-line and staggered alignment.

Figure 13. Entropy generation rate vs. diameter for the selected materials.

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Figure 14. Entropy generation rate vs. approach velocity for the selected materials.

clear that the optimum entropy generation rate moves down as the thermal conductivity of the fin material increases. Copper fin gives the best performance. If the weight of the heat sink was a constraint, the aluminum fin would be preferable. However, the optimum diameter for the selected materials is almost the same. Figure 14 shows the variation of entropy generation rate with approach velocity for design variables: inline alignment, circular pin-fin configuration, and the selected fin materials—plastic, aluminum, and copper. The optimum entropy generation rate moves down again with the thermal conductivity of the fin material, which shows the best performance of copper fins for the same approach velocity. It also shows that the optimum approach velocity for the selected materials is almost the same under the same operating conditions. Figure 15 shows the effect of number of fins on the entropy generation rate for the selected fin materials. For each material, the same optimum number of fins exist; however, the entropy generation rate decreases with the increase in thermal conductivity for the same number of fins and under the same operating conditions. The effect of the fin length on the entropy generation rate for the selected fin materials is shown in Figure 16. It is noticed that the entropy generation rate decreases up to the optimum point and then increases

Figure 15. Entropy generation rate vs. number of fins for the selected materials.

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Figure 16. Entropy generation rate vs. length of fins for the selected materials.

with the fin length. Also, the optimum fin length increases with thermal conductivity of fin materials.

7. Results Summary From the various calculations for optimization of the selected geometries and pin-fin materials in case of both in-line and staggered alignments, Table 2 shows the optimum diameter, optimum length, number of fins, and the entropy generation rate for the present study.

8. Conclusions In this work, the calculation is presented that allows design variables in pin-fin heat sink to be optimized. The comparison between the staggered and in-line alignment pinfins heat sink for the selected geometries and pin-fin materials is presented. A group of variables are taken into consideration to obtain an integral study of the heat sink such as diameter, length, velocity, number of fins, different configurations, and fin-material. Based on the results, the following conclusions are drawn: 1. The thermal performance improves with increasing number of fins and as total convective surface area increases with the fin length. 2. Elliptical and circular pin-fin geometries out-perform rhombus, rectangular, and square shaped pin-fins. 3. Staggered alignment performs better than in-line alignment. 4. In a particular pin-fin configuration, entropy generation rate is comparatively elliptical < circular < rhombus < rectangular < square pin-fins. 5. The operating conditions of the pin-fin heat sink: diameter, length, and number of fins for in-line alignment are greater than that of staggered alignment. 6. The size and corresponding material cost are more for in-line alignment. 7. Copper fin gives the best performance, and if the weight of the heat sink is a constraint, the aluminum fin would be preferable. 8. The optimum diameter for the selected materials is almost the same. 9. The optimum fin length increases with thermal conductivity of fin materials. 10. The entropy generation rate decreases with the increase in thermal conductivity for the same number of fins and under the same operating conditions.

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Z. S. Abdel-Rehim Table 2 Optimum variables for the selected geometries and pin-fin materials in both alignments of in-line and staggered

Geometries

Alignment

Circular

In-line

Staggered


Square

In-line

Staggered

Rhombus

In-line

Staggered

Rectangular

In-line

Staggered

Elliptical

In-line

Staggered

Pin-fin materials, Kth , W/mK

Optimum length, Lopt , mm

Optimum diameter, Dopt , mm

Plastic D 25 Aluminum D 237 Copper D 400 Plastic D 25 Aluminum D 237 Copper D 400 Plastic D 25 Aluminum D 237 Copper D 400 Plastic D 25 Aluminum D 237 Copper D 400 Plastic D 25 Aluminum D 237 Copper D 400 Plastic D 25 Aluminum D 237 Copper D 400 Plastic D 25 Aluminum D 237 Copper D 400 Plastic D 25 Aluminum D 237 Copper D 400 Plastic D 25 Aluminum D 237 Copper D 400 Plastic D 25 Aluminum D 237 Copper D 400

21.039 18.04 15.2 16.37 14.01 10.2 18.33 17.01 15.03 17.003 15.12 13.01 18.24 14.4 12.05 17.9 13.2 11.24 16.8 15.2 12.01 15.8 12.99 11.8 19.9 16.2 14.0 17.9 14.6 12.03

2.02 1.9 1.6 1.9 1.6 1.5 2.2 2.01 1.9 2.19 2.0 1.99 2.19 2.0 1.99 1.9 1.88 1.85 2.04 2.02 2.001 2.01 2.0 1.98 2.0 1.9 1.7 1.9 1.7 1.67

Number of pin-fins 10 10 10 9 9 9 10 10 10 9 9 9 10 10 10 9 9 9 10 10 10 9 9 9 10 10 10 9 9 9

10 10 10 9 9 9 10 10 10 9 9 9 10 10 10 9 9 9 10 10 10 9 9 9 10 10 10 9 9 9

Entropy generation rate, W/K 5.8 5.1 5.0 6.33 5.57 5.2 7.8 6.072 6.01 8.9 7.8 7.2 7.7 7.0 6.2 7.99 7.6 6.5 8.0 7.7 7.2 8.8 7.5 7.3 6.3 6.0 5.5 6.8 6.5 5.9

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

References Bahrami, M., Culham, J. R., and Yovanovich, M. M. 2003. A scale analysis approach to thermal contact resistance. ASME International Mechanical Engineering Congress and RDD Expo, Washington, DC, November 16–21. Bahrami, M., Culham, J. R., and Yovanovich, M. M. 2004. Thermal resistances of gaseous gap for conforming rough contacts. 42nd AIAA Aerospace Meeting and Exhibit, Reno, NV, January 5–8. Bejan A. 1996. Entropy Generation Minimization. New York: CRC Press. Culham, J. R., Yovanovich, M. M., and Lee, S. 1995. Thermal modeling of isothermal cuboids and rectangular heat sinks cooled by natural convection. IEEE Trans. Compon. Packag. Manufact. Technol.—Part A 18:559–566. Culham, R., and Muzychka, Y. 2001. Optimization of plate heat sinks using entropy generation minimization. IEEE Trans. Compon. Packag. Manufact. Technol. 24:159–165.


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Kondo, Y., Matsushima, H., and Komatsu, T. 2000. Optimization of pin-fin heat sinks for impingement cooling of electronic packages. J. Electron. Packag. 122:240–246. Kothandaraman, C. P., and Subramanyan, S. 1989. Heat and Mass Transfer Data Book. New Delhi, New Age International Publishers. Lee, S. 1995. Optimum design and selection of heat sinks. 1995. IEEE Trans. Compon. Packag. Manufact. Technol. 18:812–817. Morgan, V. T. 1975. The overall convective heat transfer from smooth circular cylinders. Adv. Heat Transf. 11:199–264. Perkins, H. C., and Leppert, G. 1964. Local heat transfer coefficients on a uniformly heated cylinder. Int. J. Heat Mass Transfer 7:143–158. Shaukatullah, H., Storr, W. R., Hansen, B. J., and Gaynes, M. A. 1996. Design and optimization of pin fin heat sinks for low velocity applications. IEEE Trans. Compon. Packag. Manufact. Technol—Part A 19:486–495. Shih, C. J., and Liu, G. C. 2004. Optimal design methodology of plate-fin heat sinks for electronic cooling using entropy generation strategy. IEEE Trans. Compon. Packag. Manufact. Technol. 27:551–559. Shuja, S. Z. (2002). Optimal fin geometry based on exergoeconomic analysis for a pin-fin array with application to electronic cooling. Exergy Int. J. 2:284–258.